Approximation to Stochastic Variance Reduced Gradient Langevin Dynamics by Stochastic Delay Differential Equations

Peng, Chen; Lu, Jianya; Xu, Lihu

doi:10.1007/s00245-022-09854-3

Cited by 6 publications

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“…The recent use of Langevin samplings in machine learning, has caused a surge of interest error bounds for the invariant measures of the solution of the SDE and of the EM discretization, see e.g. [22,30,3,12]. To the best of our knowledge, this paper is the first contribution studying the bound between the invariant measures of solutions to SDEs driven by stable noise and their EM discretizations.…”

Section: Introductionmentioning

confidence: 99%

Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme

Chen¹,

Deng²,

Schilling³

et al. 2022

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We study approximations of the invariant measure of a stochastic differential equation (SDE) driven by an α-stable Lévy process (1 < α < 2) using the Euler-Maruyama (EM) scheme. By a discrete version of Duhamel's principle and Bismut's formula in Malliavin calculus, we show that the Wasserstein-1 distance of the invariant measures of the solution and the approximation is -up to a constant -bounded by η 2 α −1 where η is the step size of the EM scheme. An explicit calculation for Ornstein-Uhlenbeck α-stable process shows that the rate ηexample is given, and the relevant simulations are implemented. CONTENTS 1. Introduction 1.1. Motivation, contribution and method 1.2. Notation 1.3. Assumptions and main result 2. Ergodicity and moment estimates 3. Proof of Theorem 1.2 3.1. Auxiliary lemmas 3.2. Proof of Theorem 1.2 4. Malliavin calculus and the proof of Lemma 3.1 4.1. Jacobi flow associated with the SDE (1.1) 4.2. Bismut's formula 4.3. Time-change method for the SDE (1.1) 4.4. Proof of Lemma 3.1 5. Example and numerical simulations Appendix A. Proofs of Propositions 2.1, 2.2 and 2.3 and Lemma 2.5 Appendix B. Exact rate for the Ornstein-Uhlenbeck process References

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